To catch rabbits, chickens and squirrels right click on the animal and you will automatically pursue it, once you have caught up to it it will appear in your inventory. For rabbits and squirrels you will need to use the second speed on your character, run speed.
Note that you need to have the available space in your inventory, rabbits and chickens take four slots, 2x2 and squirrels take up two spaces. (vertically)
Once you have caught it, right click it again in your inventory to bring up the 'wring neck', option to slaughter it.

I've included the links to the wiki and Sevenless' beginner guide which although both are outdated in places will still give you lots of help when starting out.
http://ringofbrodgar.com/wiki/Haven_and_Hearth_WikiSevenless' beginner guide
Koru wrote:It is like in Lord of the Flies, nobody controlls what is going on in the hearthlands, those weaker and with conscience are just fucked.
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